L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (0.965 + 0.258i)5-s + (0.258 − 0.965i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s − 1.00i·9-s + (−0.965 + 0.258i)10-s + (0.258 + 0.965i)12-s + (1.22 + 0.707i)13-s + (−0.499 + 0.866i)14-s + (−0.866 + 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s + 0.517·19-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (0.965 + 0.258i)5-s + (0.258 − 0.965i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s − 1.00i·9-s + (−0.965 + 0.258i)10-s + (0.258 + 0.965i)12-s + (1.22 + 0.707i)13-s + (−0.499 + 0.866i)14-s + (−0.866 + 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s + 0.517·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8968825876\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8968825876\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.965 - 0.258i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 0.517T + T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.210941628265493328929074396418, −8.611612988160453906142389892325, −7.75466315218741579834870657777, −6.67815691251148048973292909695, −6.27003997348337872234126665172, −5.48618834714801229239884378871, −4.72924604149806789351937969806, −3.71736458147342043039422711240, −2.14024758033673906357569617643, −1.14900841890967768459374858563,
1.17084235097208444509567113522, 1.80908979797878936847769555546, 2.77463542601717647382404353591, 4.13794906804047909800570970391, 5.44453923860002821342851477174, 5.82764455482139827662304924212, 6.71114960200923401479929273623, 7.69880433052784916615484063118, 8.245397941052528534429490066287, 8.885998025782073653136813467270